The Noise Collector for sparse recovery in high dimensions

Area of Science:

• Signal Processing
• High-Dimensional Data Analysis
• Sparse Signal Recovery

Background:

• Detecting sparse signals in noisy, high-dimensional data is crucial in science and engineering.
• Traditional L1-norm minimization for sparse solutions requires optimal regularization parameter selection, which is challenging with noisy data.

Purpose of the Study:

• To develop an efficient sparse signal detection method that eliminates the need for parameter estimation.
• To achieve zero false discovery rate (FDR) in the presence of noise.

Main Methods:

• Introduced a no-phantom weight (τ) and Noise Collector matrix (C).
• Solved an augmented system by incorporating noise (e) into the original linear system.
• Utilized L1-norm minimization on the augmented system.

Main Results:

• The L1-norm minimal solution of the augmented system achieves a zero false discovery rate for any noise level, with high probability as data dimension increases.
• Exact support recovery is possible when noise levels are not excessive.
• A fast Noise Collector algorithm was developed, making the augmented system's computational cost comparable to the original system.

Conclusions:

• The proposed method offers a robust and efficient solution for sparse signal detection in noisy environments.
• The technique demonstrates practical effectiveness, as shown in passive array imaging applications.
• This approach advances signal processing capabilities for complex, real-world data.